[Patent Reference Document 1] Japanese Patent Publication No.2000-231012, “Method for Designing a Diffractive Optical Element”
This is one of the former inventions made by the inventor of the present invention. The application proposed a diffractive optical elements (DOE: Fraunhofer type) having functions of dividing a laser beam. The invention aimed at facilitating the designing of DOEs by assigning inherently narrow tolerance parameters with larger tolerances enlarged by the estimating designed DOEs by a merit function including influences of errors for the selected parameters. Enlargement of the tolerances makes the production of the DOEs easy.
The DOE is a set of lengthwise and crosswise aligning identical unit patterns. A unit pattern is a set of lengthwise and crosswise aligning pixels which are the smallest cell having a parameter. Since all the unit patterns are identical, the design of a DOE can be reduced to design of a unit pattern. The DOE with unit patterns diffracts a beam into a plurality of divided beams just on lattice points assumed on an image plane. The Fast Fourier Transformation (FFT) algorithm can calculate angular intensity distribution of diffracted beams separated by a DOE.
[Patent Reference Document 2] Japanese Patent No.3,346,374, “Laser Boring Apparatus”
This is another of the former inventions made by the inventor of the present invention. The patent No. 3,346,374 proposed a laser boring apparatus for boring many holes with the same diameter at the same interval simultaneously on an object plane by dividing a laser beam into many identical separated beams and converging the separated beams by an f sin θ lens in parallel on lattice points assumed on the object plane. Coupling of a DOE (Fraunhofer type or Fresnel type) and an f sin θ lens raises position accuracy of the hole boring by the laser beams. The patent allows the apparatus to utilize higher order diffraction beams effectively without loss. The patented invention succeeds in boring many holes at a stroke and has various advantages.
In the prior invention, the DOE was defined as an optical part which includes spatial repetitions of identical unit patterns  with a definite spatial period (interval) Λ and diffracts a laser beam into many divided beams. The DOE is made by designing a unit pattern , aligning H identical unit patterns  in succession in a horizontal (x-) direction, and aligning G identical unit patterns  in a vertical (y-) direction.
Then design of the DOE was equivalent to design of the unit pattern . The DOE has a set of vertically-aligning R pixels and horizontally-aligning S pixels. A pixel is a minimum part whose thickness (heights) can take one of g=2s (s: integer) different values. The full span of thickness difference of the pixels corresponds to a wavelength λ of a laser beam. Pixels C are numbered by a horizontal number m and a vertical number n (0≦m≦H, 0≦n≦G) as C{(m, n)}. Determination of a set of thicknesses {dmn} of 2s different values for all the pixels C{(m, n)} was design of a DOE.
In practice, the design of the DOE was far more simplified. In the prior DOE, all the pixels could not take a arbitrary value of thicknesses but were ruled by a restriction condition. The restriction is the existence of a unit pattern . The DOE had been designed by determining thicknesses {dmn} of a set of pixels {(m, n)} included in the unit pattern  and by arranging repeatedly G×H identical unit patterns in x- and y-directions. The design of the DOE with very high degree of freedom could be replaced by the design of a unit pattern with low degree of freedom.
P x-direction aligning pixels and Q y-direction aligning pixels compose a unit pattern . The unit pattern contains PQ pixels. The number of pixels as an object of design is not the total number GHPQ of total pixels but the number PQ of pixels within a unit pattern.
A DOE has G x-aligning unit patterns and H y-aligning unit patterns. A unit pattern contains P x-aligning pixels and Q y-aligning pixels. A DOE has two-storied hierarchy. Parameters are defined here as follows.    Pixel C width (x-direction size)=a, length (y-direction size)=b.    Unit pattern  P×Q pixels, width (x-size)=Pa, length (y-size)=Qb.    DOE G×H unit patterns, width (x-size)=GPa, length (y-size)=HQb.    DOE R×S pixels (R=PG, S=HQ)    m, n coordinates of a pixel C in a unit pattern in the prior art    dmn thickness of a C(m, n) pixel taking one of g=2s values.    tmn complex transmittance of a C(m, n) pixel.    m, n coordinates of a pixel C in the whole DOE in the present invention.    p, q diffraction order parameters (integers) of the prior art    α, β diffraction angles (not integers; continual) of the present invention.
The set of notations is used throughout this description. A DOE includes RS (=PGHQ) pixels. A pixel takes one of the g=2s (s: integer) different thicknesses. The total degree of freedom of a DOE is gRS(=2sPGHQ). But, actual degree of freedom of a prior art DOE is only gPQ, since repetitions of identical unit patterns compose a prior art DOE. The practical degree of freedom for designing is equal to gPQ which is a quotient of the total freedom degree gPGHQ divided by the unit pattern number GH. What is important is the number of pixels which should be determined for designing a DOE. The pixel thickness parameter freedom g is less important here. Then, the degree of freedom for designing a DOE will signify the number of outstanding pixels (PGHQ or PQ) hereafter.
For example, assuming that a unit pattern  includes 1000 pixels (PQ=1000) and a DOE includes 1000 unit patterns (GH=1000), the DOE includes 1000000 pixels (PQGH=1000000). However, it is unnecessary to determine thicknesses of all the 1000000 pixels. Instead, DOE designing should determine only the thicknesses of 1000 pixels. Free parameters are greatly reduced to one thousandth.
Existence of repeating unit patterns alleviates the quantity of calculation to a great extent. The repetition of identical unit patterns gives a DOE a role of a diffraction grating which allows divided light to focus only at lattice points defined on an image plane. Extra regions except lattice points are all dark. The unit patterned DOE makes only beam spots only on lattice points which are distributed at predetermined spatial intervals (periods) on an image plane. The lattice point diffraction enables the Fast Fourier Transformation (FFT) to calculate the intensity of diffracted beams at lattice points. Use of the FFT has exquisite advantages of fast calculation of a diffracted image and facile designing of the DOE.
Furthermore, hole-boring processing, in many cases, bores many holes at lattice points aligning lengthwise and crosswise at a common interval. For example, print circuit boards sometimes require that small holes are bored to be regularly arranged at a spatial period in two dimensions.
Boring holes at lattice points fortunately corresponds to Fourier transformation of repetition of identical unit patterns. The FFT is a suitable calculation of the lattice point boring. Strictly speaking, since sine functions (sin θ) of diffraction angles θ are multiples of a unit value, an f sin θ lens instead of an ordinary f tan θ lens allows the DOE to make lattice point diffraction. The lattice point diffraction can be automatically and rapidly calculated by the Fast Fourier Transformation (FFT).